Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1024,2,Mod(257,1024)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1024.257"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1024, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1024 = 2^{10} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1024.e (of order \(4\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,-4,0,0,0,0,0,0,0,4,0,0,0,0,0,-12,0,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(8.17668116698\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 128)
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label 769.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1024.769
Dual form 1024.2.e.a.257.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-2.00000 + 2.00000i) q^{3} -5.00000i q^{9} +(2.00000 + 2.00000i) q^{11} -6.00000 q^{17} +(-6.00000 + 6.00000i) q^{19} -5.00000i q^{25} +(4.00000 + 4.00000i) q^{27} -8.00000 q^{33} -6.00000i q^{41} +(-6.00000 - 6.00000i) q^{43} +7.00000 q^{49} +(12.0000 - 12.0000i) q^{51} -24.0000i q^{57} +(-10.0000 - 10.0000i) q^{59} +(6.00000 - 6.00000i) q^{67} -2.00000i q^{73} +(10.0000 + 10.0000i) q^{75} -1.00000 q^{81} +(2.00000 - 2.00000i) q^{83} +18.0000i q^{89} -10.0000 q^{97} +(10.0000 - 10.0000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{3} + 4 q^{11} - 12 q^{17} - 12 q^{19} + 8 q^{27} - 16 q^{33} - 12 q^{43} + 14 q^{49} + 24 q^{51} - 20 q^{59} + 12 q^{67} + 20 q^{75} - 2 q^{81} + 4 q^{83} - 20 q^{97} + 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1024\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(1023\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −2.00000 + 2.00000i −1.15470 + 1.15470i −0.169102 + 0.985599i \(0.554087\pi\)
−0.985599 + 0.169102i \(0.945913\pi\)
\(4\) 0 0
\(5\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 0 0
\(9\) − 5.00000i − 1.66667i
\(10\) 0 0
\(11\) 2.00000 + 2.00000i 0.603023 + 0.603023i 0.941113 − 0.338091i \(-0.109781\pi\)
−0.338091 + 0.941113i \(0.609781\pi\)
\(12\) 0 0
\(13\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −6.00000 −1.45521 −0.727607 − 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) 0 0
\(19\) −6.00000 + 6.00000i −1.37649 + 1.37649i −0.526026 + 0.850469i \(0.676318\pi\)
−0.850469 + 0.526026i \(0.823682\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) − 5.00000i − 1.00000i
\(26\) 0 0
\(27\) 4.00000 + 4.00000i 0.769800 + 0.769800i
\(28\) 0 0
\(29\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(30\) 0 0
\(31\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) −8.00000 −1.39262
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 6.00000i − 0.937043i −0.883452 − 0.468521i \(-0.844787\pi\)
0.883452 − 0.468521i \(-0.155213\pi\)
\(42\) 0 0
\(43\) −6.00000 − 6.00000i −0.914991 − 0.914991i 0.0816682 − 0.996660i \(-0.473975\pi\)
−0.996660 + 0.0816682i \(0.973975\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 7.00000 1.00000
\(50\) 0 0
\(51\) 12.0000 − 12.0000i 1.68034 − 1.68034i
\(52\) 0 0
\(53\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 24.0000i − 3.17888i
\(58\) 0 0
\(59\) −10.0000 − 10.0000i −1.30189 − 1.30189i −0.927117 − 0.374772i \(-0.877721\pi\)
−0.374772 − 0.927117i \(-0.622279\pi\)
\(60\) 0 0
\(61\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 6.00000 − 6.00000i 0.733017 − 0.733017i −0.238200 − 0.971216i \(-0.576557\pi\)
0.971216 + 0.238200i \(0.0765572\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) − 2.00000i − 0.234082i −0.993127 − 0.117041i \(-0.962659\pi\)
0.993127 − 0.117041i \(-0.0373409\pi\)
\(74\) 0 0
\(75\) 10.0000 + 10.0000i 1.15470 + 1.15470i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) −1.00000 −0.111111
\(82\) 0 0
\(83\) 2.00000 − 2.00000i 0.219529 − 0.219529i −0.588771 − 0.808300i \(-0.700388\pi\)
0.808300 + 0.588771i \(0.200388\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 18.0000i 1.90800i 0.299813 + 0.953998i \(0.403076\pi\)
−0.299813 + 0.953998i \(0.596924\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −10.0000 −1.01535 −0.507673 − 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) 0 0
\(99\) 10.0000 − 10.0000i 1.00504 − 1.00504i
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1024.2.e.a.769.1 2
4.3 odd 2 1024.2.e.f.769.1 2
8.3 odd 2 CM 1024.2.e.a.769.1 2
8.5 even 2 1024.2.e.f.769.1 2
16.3 odd 4 1024.2.e.f.257.1 2
16.5 even 4 1024.2.e.f.257.1 2
16.11 odd 4 inner 1024.2.e.a.257.1 2
16.13 even 4 inner 1024.2.e.a.257.1 2
32.3 odd 8 256.2.a.e.1.2 2
32.5 even 8 128.2.b.a.65.1 ✓ 2
32.11 odd 8 128.2.b.a.65.1 ✓ 2
32.13 even 8 256.2.a.e.1.2 2
32.19 odd 8 256.2.a.e.1.1 2
32.21 even 8 128.2.b.a.65.2 yes 2
32.27 odd 8 128.2.b.a.65.2 yes 2
32.29 even 8 256.2.a.e.1.1 2
96.5 odd 8 1152.2.d.c.577.2 2
96.11 even 8 1152.2.d.c.577.2 2
96.29 odd 8 2304.2.a.t.1.1 2
96.35 even 8 2304.2.a.t.1.2 2
96.53 odd 8 1152.2.d.c.577.1 2
96.59 even 8 1152.2.d.c.577.1 2
96.77 odd 8 2304.2.a.t.1.2 2
96.83 even 8 2304.2.a.t.1.1 2
160.19 odd 8 6400.2.a.by.1.2 2
160.27 even 8 3200.2.f.o.449.4 4
160.29 even 8 6400.2.a.by.1.2 2
160.37 odd 8 3200.2.f.o.449.1 4
160.43 even 8 3200.2.f.o.449.3 4
160.53 odd 8 3200.2.f.o.449.2 4
160.59 odd 8 3200.2.d.c.1601.1 2
160.69 even 8 3200.2.d.c.1601.2 2
160.99 odd 8 6400.2.a.by.1.1 2
160.107 even 8 3200.2.f.o.449.1 4
160.109 even 8 6400.2.a.by.1.1 2
160.117 odd 8 3200.2.f.o.449.4 4
160.123 even 8 3200.2.f.o.449.2 4
160.133 odd 8 3200.2.f.o.449.3 4
160.139 odd 8 3200.2.d.c.1601.2 2
160.149 even 8 3200.2.d.c.1601.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
128.2.b.a.65.1 ✓ 2 32.5 even 8
128.2.b.a.65.1 ✓ 2 32.11 odd 8
128.2.b.a.65.2 yes 2 32.21 even 8
128.2.b.a.65.2 yes 2 32.27 odd 8
256.2.a.e.1.1 2 32.19 odd 8
256.2.a.e.1.1 2 32.29 even 8
256.2.a.e.1.2 2 32.3 odd 8
256.2.a.e.1.2 2 32.13 even 8
1024.2.e.a.257.1 2 16.11 odd 4 inner
1024.2.e.a.257.1 2 16.13 even 4 inner
1024.2.e.a.769.1 2 1.1 even 1 trivial
1024.2.e.a.769.1 2 8.3 odd 2 CM
1024.2.e.f.257.1 2 16.3 odd 4
1024.2.e.f.257.1 2 16.5 even 4
1024.2.e.f.769.1 2 4.3 odd 2
1024.2.e.f.769.1 2 8.5 even 2
1152.2.d.c.577.1 2 96.53 odd 8
1152.2.d.c.577.1 2 96.59 even 8
1152.2.d.c.577.2 2 96.5 odd 8
1152.2.d.c.577.2 2 96.11 even 8
2304.2.a.t.1.1 2 96.29 odd 8
2304.2.a.t.1.1 2 96.83 even 8
2304.2.a.t.1.2 2 96.35 even 8
2304.2.a.t.1.2 2 96.77 odd 8
3200.2.d.c.1601.1 2 160.59 odd 8
3200.2.d.c.1601.1 2 160.149 even 8
3200.2.d.c.1601.2 2 160.69 even 8
3200.2.d.c.1601.2 2 160.139 odd 8
3200.2.f.o.449.1 4 160.37 odd 8
3200.2.f.o.449.1 4 160.107 even 8
3200.2.f.o.449.2 4 160.53 odd 8
3200.2.f.o.449.2 4 160.123 even 8
3200.2.f.o.449.3 4 160.43 even 8
3200.2.f.o.449.3 4 160.133 odd 8
3200.2.f.o.449.4 4 160.27 even 8
3200.2.f.o.449.4 4 160.117 odd 8
6400.2.a.by.1.1 2 160.99 odd 8
6400.2.a.by.1.1 2 160.109 even 8
6400.2.a.by.1.2 2 160.19 odd 8
6400.2.a.by.1.2 2 160.29 even 8